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I wondered what the smallest prime of the form $2n^n+k$ is for some odd $k$. For $k<91$, there are small primes, but for $k=91$ , the smallest prime (if it exists) must be very large.

  • What is the smallest prime of the form $2n^n+91$ ?

It is clear that $\gcd(91,n)=1$ must hold. $2n^n+91$ is composite for every natural number $n$ below $1000$.

$$2\times 15^{15}+91 = 42846499\times 20440124659$$

shows that the smallest prime factor can be large.

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  • $\begingroup$ No primes up to $n=1200$. Watch this space... $\endgroup$ – TonyK May 31 '15 at 11:43
  • $\begingroup$ Did you also check the range $0\le n \le 1000$ ? $\endgroup$ – Peter May 31 '15 at 11:46
  • $\begingroup$ Yes.${}{}{}{}{}{}$ $\endgroup$ – TonyK May 31 '15 at 11:47
  • $\begingroup$ No primes up to $n=1500$... $\endgroup$ – TonyK May 31 '15 at 11:53
  • $\begingroup$ I experience a deja-vu... There must be a glitch in the Matrix ! $\endgroup$ – Lucian May 31 '15 at 12:35
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$$2 \times 1949^{1949} + 91$$

is probably prime! (Running a rigorous primality test on it would take almost a whole day $-$ see here, for instance $-$ so I'm not going to do that.) $2n^n+91$ is composite for all lower values of $n$.

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  • $\begingroup$ How did you conclude that it is 'probably prime'? But it does look suspiciously prime... $\endgroup$ – Trogdor May 31 '15 at 12:45
  • $\begingroup$ @Trogdor there are plenty of test that show a number is composite, if they fail it is "probably prime." See en.wikipedia.org/wiki/Probable_prime $\endgroup$ – quid May 31 '15 at 12:59
  • $\begingroup$ I like "suspiciously prime". All we have to do is decide on a meaning for it. $\endgroup$ – TonyK May 31 '15 at 13:05
  • $\begingroup$ You're the one who made me recant my skeptical statements about proving a large number prime. Same person asking, too. They deleted my revised answer, which is a shame. math.stackexchange.com/questions/402357/… $\endgroup$ – Will Jagy May 31 '15 at 15:32
  • $\begingroup$ @WillJagy: Philistines. $\endgroup$ – TonyK May 31 '15 at 16:18

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